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The hyperbolic cotangent remainder

Abstract

A Bernoulli remainder and two consecutive derivative signs.

Definition 1.1 (The remainder).

Lean statement: D5/S3/Analytic/CothBernoulliDegree/Defs.p

Formalization. D5/S3/Analytic/CothBernoulliDegree/Defs.p (✓ std3).

Citation. L. Zhu (2020). Completely monotonic integer degrees for a class of special functions. DOI: 10.3934/math.2020224. URL: https://www.aimspress.com/article/doi/10.3934/math.2020224.

Commentary.

For a natural n, subtract from cosh(t)/sinh(t)/t the sum over k from zero through n of 2^(2k) B_(2k) t^(2k-2)/(2k)!. The powers of t in this finite sum are integer powers. Only even Bernoulli indices occur.

Definition 1.2 (The weighted remainder).

Lean statement: D5/S3/Analytic/CothBernoulliDegree/Defs.P

Formalization. D5/S3/Analytic/CothBernoulliDegree/Defs.P (✓ std3).

Citation. L. Zhu (2020). Completely monotonic integer degrees for a class of special functions. DOI: 10.3934/math.2020224. URL: https://www.aimspress.com/article/doi/10.3934/math.2020224.

Commentary.

For natural m and n, P(m,n,t) is (-1)^n times t^m times the remainder p(n,t).

Definition 1.3 (Two consecutive derivative orders).

Lean statement: D5/S3/Analytic/CothBernoulliDegree/Defs.claim

Formalization. D5/S3/Analytic/CothBernoulliDegree/Defs.claim (✓ std3).

Citation. L. Zhu (2020). Completely monotonic integer degrees for a class of special functions. DOI: 10.3934/math.2020224. URL: https://www.aimspress.com/article/doi/10.3934/math.2020224.

Commentary.

For every natural m at least one and natural n at least two, the derivative of P(m,n) of order m+2n-2 is strictly positive at every positive real t. The derivative of order m+2n-1 is not strictly positive at every positive real t.

References